Browsing by Subject "convex function"

Now showing 1 - 6 of 6

Access status: Open Access ,
A new characterization of convex φ-functions with a parameter
(2015) Micherda, Bartosz
We show that, under some additional assumptions, all projection operators onto latticially closed subsets of the Orlicz-Musielak space generated by $\Phi$ are isotonic if and only if $\Phi$ is convex with respect to its second variable. A dual result of this type is also proven for antiprojections. This gives the positive answer to the problem presented in Opuscula Mathematica in 2012.
Access status: Open Access ,
Conjugate functions, Lp-norm like functionals, the generalized Hölder inequality, Minkowski inequality and subhomogeneity
(2014) Matkowski, Janusz
For $h:(0,\infty )\rightarrow \mathbb{R}$, the function $h^{\ast }\left( t\right) :=th(\frac{1}{t})$ is called $(∗)$-conjugate to $h$. This conjugacy is related to the Hölder and Minkowski inequalities. Several properties of $(∗)$-conjugacy are proved. If $\varphi$ and $\varphi ^{\ast }$ are bijections of $\left(0,\infty \right)$ then $(\varphi ^{-1}) ^{\ast }=\left( \left[ \left( \varphi ^{\ast }\right) ^{-1}\right] ^{\ast }\right) ^{-1}$. Under some natural rate of growth conditions at $0$ and $infty$, if $\varphi$ is increasing, convex, geometrically convex, then $\left[ \left( \varphi^{-1}\right) ^{\ast }\right] ^{-1}$ has the same properties. We show that the Young conjugate functions do not have this property. For a measure space $(\Omega ,\Sigma ,\mu )$ denote by $S=S(\Omega ,\Sigma ,\mu )$ the space of all $\mu$-integrable simple functions $x:\Omega \rightarrow \mathbb{R}$. Given a bijection $\varphi :(0,\infty )\rightarrow (0,\infty )$, define $\mathbf{P}_{\varphi }:S\rightarrow \lbrack 0,\infty )$ by $\mathbf{P}_{\varphi }(x):=\varphi ^{-1}\bigg( \int\limits_{\Omega (x)}\varphi \circ \left\vert x\right\vert d\mu \bigg),$ where $\Omega(x)$ is the support of $x$. Applying some properties of the $(∗)$ operation, we prove that if $\int\limits_{\Omega }xy\leq \mathbf{P}_{\varphi }(x)\mathbf{P}_{\psi }(y)$ where $\varphi ^{-1}$ and $\psi ^{-1}$ are conjugate, then $\varphi$ and $\psi$ are conjugate power functions. The existence of nonpower bijections $\varphi$ and $\psi$ with conjugate inverse functions $\psi =\left[ ( \varphi ^{-1}) ^{\ast}\right] ^{-1}$ such that $\mathbf{P}_{\varphi }$ and $\mathbf{P}_{\psi }$ are subadditive and subhomogeneous is considered.
Access status: Restricted ,
Funkcje wypukłe i nierówności z nimi związane
(Data obrony: 2010-07-08) Ostachowski, Piotr
Wydział Matematyki Stosowanej
Access status: Restricted ,
Miary podwójnie stochastyczne o zadanym nośniku i równania funkcyjne
(Data obrony: 2009-06-17) Bazarnik, Krzysztof
Wydział Matematyki Stosowanej
Access status: Open Access ,
On some inequality of Hermite-Hadamard type
(2012) Wąsowicz, Szymon; Witkowski, Alfred
It is well-known that the left term of the classical Hermite–Hadamard inequality is closer to the integral mean value than the right one. We show that in the multivariate case it is not true. Moreover, we introduce some related inequality comparing the methods of the approximate integration, which is optimal. We also present its counterpart of Fejér type.
Access status: Open Access ,
Reiterated periodic homogenization of integral functionals with convex and nonstandard growth integrands
(Wydawnictwa AGH, 2021) Tachago, Joel Fotso; Nnang, Hubert; Zappale, Elvira
Multiscale periodic homogenization is extended to an Orlicz-Sobolev setting. It is shown by the reiteraded periodic two-scale convergence method that the sequence of minimizers of a class of highly oscillatory minimizations problems involving convex functionals, converges to the minimizers of a homogenized problem with a suitable convex function.